Idea Transcript
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6
M1
GEOMETRY
Lesson 6: Solve for Unknown Angles—Angles and Lines at a
tilll
Point Student Outcomes
Students review formerly learned geometry facts and practice citing the geometric justifications in anticipation of unknown angle proofs.
Lesson Notes Lessons 1–5 serve as a foundation for the main subject of this module, which is congruence. By the end of the unknown angles lessons (Lessons 6–8), students start to develop fluency in two areas: (1) solving for unknown angles in diagrams and (2) justifying each step or decision in the proof-writing process of unknown angle solutions. The “missing-angle problems” in this topic occur in many American geometry courses and play a central role in some Asian curricula. A missing-angle problem asks students to use several geometric facts together to find angle measures in a diagram. While the simpler problems require good, purposeful recall and application of geometric facts, some problems are complex and may require ingenuity to solve. Historically, many geometry courses have not expected this level of sophistication. Such courses would not have demanded that students use their knowledge constructively but rather to merely regurgitate information. The missing-angle problems are a step up in problem solving. Why do we include them at this juncture in this course? The main focal points of these opening lessons are to recall or refresh and supplement existing conceptual vocabulary, to emphasize that work in geometry involves reasoned explanations, and to provide situations and settings that support the need for reasoned explanations and that illustrate the satisfaction of building such arguments. Lesson 6 is Problem Set based and focuses on solving for unknown angles in diagrams of angles and lines at a point. By the next lesson, students should be comfortable solving for unknown angles numerically or algebraically in diagrams involving supplementary angles, complementary angles, vertical angles, and adjacent angles at a point. As always, vocabulary is critical, and students should be able to define the relevant terms themselves. It may be useful to draw or discuss counterexamples of a few terms and ask students to explain why they do not fit a particular definition. As students work on problems, encourage them to show each step of their work and to list the geometric reason for each step (e.g., “Vertical angles have equal measure”). This prepares students to write a reason for each step of their unknown angle proofs in a few days. A chart of common facts and discoveries from middle school that may be useful for student review or supplementary instruction is included at the end of this lesson. The chart includes abbreviations students may have previously seen in middle school, as well as more widely recognized ways of stating these ideas in proofs and exercises.
Lesson 6:
Solve for Unknown Angles—Angles and Lines at a Point
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Classwork Opening Exercise (5 minutes) Ask students to find the missing angles in these diagrams. The exercise reminds students of the basics of determining missing angles that they learned in middle school. Discuss the facts that students recall, and use these as a starting point for the lesson. Opening Exercise Determine the measure of the missing angle in each diagram.
What facts about angles did you use? Answers may include the following: vertical angles are equal in measure; linear pairs form supplementary angles; angles at a point sum to 𝟑𝟔𝟎°.
Discussion (4 minutes) Discussion
MP.6
Two angles ∠𝑨𝑶𝑪 and ∠𝑪𝑶𝑩, with a common side ⃗⃗⃗⃗⃗⃗ 𝑶𝑪, are adjacent angles if 𝑪 belongs to the interior of ∠𝑨𝑶𝑩. The sum of angles on a straight line is 𝟏𝟖𝟎°, and two such angles are called a linear pair. Two angles are called supplementary if the sum of their measures is 𝟏𝟖𝟎° ; two angles are called complementary if the sum of their measures is 𝟗𝟎° . Describing angles as supplementary or complementary refers only to the measures of their angles. The positions of the angles or whether the pair of angles is adjacent to each other is not part of the definition. In the figure, line segment 𝑨𝑫 is drawn. Find 𝒎∠𝑫𝑪𝑬. 𝒎∠𝑫𝑪𝑬 = 𝟏𝟖°
The total measure of adjacent angles around a point is 𝟑𝟔𝟎°
Lesson 6:
.
Solve for Unknown Angles—Angles and Lines at a Point
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Find the measure of ∠𝑯𝑲𝑰.
𝐻
𝒎∠𝑯𝑲𝑰 = 𝟖𝟎°
𝐾
𝐺
𝐼 Vertical angles have
equal measure. Two angles are vertical if their sides form opposite rays.
Find 𝒎∠𝑻𝑹𝑽.
𝑇
𝑆
𝒎∠𝑻𝑹𝑽 = 𝟓𝟐° 𝑅
𝑈
𝑉
Example (5 minutes) Example Find the measure of each labeled angle. Give a reason for your solution.
Angle
Angle Measure
Reason
∠𝒂
𝟑𝟓°
Linear pairs form supplementary angles.
∠𝒃
𝟏𝟒𝟎°
Linear pairs form supplementary angles.
∠𝒄
𝟒𝟎°
Vertical angles are equal in measure.
∠𝒅
𝟏𝟒𝟎°
Linear pairs form supplementary angles, or vertical angles are equal in measure.
∠𝒆
𝟗𝟕°
Angles at a point sum to 𝟑𝟔𝟎°.
Lesson 6:
Solve for Unknown Angles—Angles and Lines at a Point
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Exercises (25 minutes) Exercises
MP.7
̅̅̅̅, and 𝑬𝑭 ̅̅̅̅, 𝑪𝑫 ̅̅̅̅ are straight line segments. Find the measure of each marked angle, or find the In the figures below, 𝑨𝑩 unknown numbers labeled by the variables in the diagrams. Give reasons for your calculations. Show all the steps to your solutions. 1.
𝒎∠𝒂 = 𝟑𝟔° Linear pairs form supplementary angles.
2.
𝒎∠𝒃 = 𝟒𝟕° Consecutive adjacent angles on a line sum to 𝟏𝟖𝟎°.
3.
𝒎∠𝒄 = 𝟏𝟒° Consecutive adjacent angles on a line sum to 𝟏𝟖𝟎°
4.
𝒎∠𝒅 = 𝟒𝟗° Consecutive adjacent angles on a line sum to 𝟏𝟖𝟎°
5. 𝒎∠𝒈 = 𝟐𝟗° Angles at a point sum to 𝟑𝟔𝟎°.
Lesson 6:
Solve for Unknown Angles—Angles and Lines at a Point
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
For Exercises 6–12, find the values of 𝒙 and 𝒚. Show all work. 6.
𝒙 = 𝟖𝟎 Angles at a point sum to 𝟑𝟔𝟎°.
7. 𝒙 = 𝟑𝟎; 𝒚 = 𝟗𝟎 Vertical angles are equal in measure. Angles at a point sum to 𝟑𝟔𝟎°.
8.
𝒙 = 𝟐𝟎 Vertical angles are equal in measure. Consecutive adjacent angles on a line sum to 𝟏𝟖𝟎°.
9.
MP.7
𝒙 = 𝟑𝟗; 𝒚 = 𝟏𝟐𝟑 Vertical angles are equal in measure. Angles at a point sum to 𝟑𝟔𝟎°.
10.
𝒙 = 𝟖𝟎; 𝒚 = 𝟏𝟐𝟐 Consecutive adjacent angles on a line sum to 𝟏𝟖𝟎°.
Lesson 6:
Solve for Unknown Angles—Angles and Lines at a Point
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
11. 𝒙 = 𝟏𝟎; 𝒚 = 𝟏𝟏𝟐 Consecutive adjacent angles on a line sum to 𝟏𝟖𝟎°. Vertical angles are equal in measure.
12.
MP.7
𝒙 = 𝟐𝟕; 𝒚 = 𝟒𝟕 Consecutive adjacent angles on a line sum to 𝟏𝟖𝟎°. Vertical angles are equal in measure.
Relevant Vocabulary Relevant Vocabulary STRAIGHT ANGLE: If two rays with the same vertex are distinct and collinear, then the rays form a line called a straight angle. VERTICAL ANGLES: Two angles are vertical angles (or vertically opposite angles) if their sides form two pairs of opposite rays.
Closing (1 minute)
What is the distinction between complementary and supplementary angles?
If two angles are adjacent and on a straight line, what are they called?
Two angles are called complementary if their angle sum is 90°, and two angles are called supplementary if their angle sum is 180°. The angles do not have to be adjacent to be complementary or supplementary. A linear pair
What is the angle sum of adjacent angles around a point?
The angle sum of adjacent angles around a point is 360°.
Exit Ticket (5 minutes)
Lesson 6:
Solve for Unknown Angles—Angles and Lines at a Point
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Name
Date
Lesson 6: Solve for Unknown Angles—Angles and Lines at a Point Exit Ticket Use the following diagram to answer the questions below:
1.
2.
a.
Name an angle supplementary to ∠𝐻𝑍𝐽, and provide the reason for your calculation.
b.
Name an angle complementary to ∠𝐻𝑍𝐽, and provide the reason for your calculation.
If 𝑚∠𝐻𝑍𝐽 = 38°, what is the measure of each of the following angles? Provide reasons for your calculations. a.
𝑚∠𝐹𝑍𝐺
b.
𝑚∠𝐻𝑍𝐺
c.
𝑚∠𝐴𝑍𝐽
Lesson 6:
Solve for Unknown Angles—Angles and Lines at a Point
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Exit Ticket Sample Solutions Use the following diagram to answer the questions below:
1. a.
Name an angle supplementary to ∠𝑯𝒁𝑱, and provide the reason for your calculation. ∠𝑱𝒁𝑭 or ∠𝑯𝒁𝑮 Linear pairs form supplementary angles.
b.
Name an angle complementary to ∠𝑯𝒁𝑱, and provide the reason for your calculation. ∠𝑱𝒁𝑨 The angles sum to 𝟗𝟎°.
2.
If 𝒎∠𝑯𝒁𝑱 = 𝟑𝟖°, what is the measure of each of the following angles? Provide reasons for your calculations. a.
𝒎∠𝑭𝒁𝑮 𝟑𝟖°
b.
𝒎∠𝑯𝒁𝑮 𝟏𝟒𝟐°
c.
𝒎∠𝑨𝒁𝑱 𝟓𝟐°
Lesson 6:
Solve for Unknown Angles—Angles and Lines at a Point
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Problem Set Sample Solutions In the figures below, ̅̅̅̅ 𝑨𝑩 and ̅̅̅̅ 𝑪𝑫 are straight line segments. Find the value of 𝒙 and/or 𝒚 in each diagram below. Show all the steps to your solutions, and give reasons for your calculations. 1. 𝒙 = 𝟏𝟑𝟑 𝒚 = 𝟒𝟑 Angle addition postulate Linear pairs form supplementary angles.
2.
𝒙 = 𝟕𝟖 Consecutive adjacent angles on a line sum to 𝟏𝟖𝟎°.
3.
𝒙 = 𝟐𝟎 𝒚 = 𝟏𝟏𝟎 Consecutive adjacent angles on a line sum to 𝟏𝟖𝟎°. Vertical angles are equal in measure.
Lesson 6:
Solve for Unknown Angles—Angles and Lines at a Point
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015
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Lesson 6
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Key Facts and Discoveries from Earlier Grades Facts (With Abbreviations Used in Grades 4–9)
Diagram/Example
Vertical angles are equal in measure.
How to State as a Reason in an Exercise or a Proof
“Vertical angles are equal in measure.”
(vert. s) 𝑎° = 𝑏°
If 𝐶 is a point in the interior of ∠𝐴𝑂𝐵, then 𝑚∠𝐴𝑂𝐶 + 𝑚∠𝐶𝑂𝐵 = 𝑚∠𝐴𝑂𝐵.
“Angle addition postulate”
(s add) 𝑚∠𝐴𝑂𝐵 = 𝑚∠𝐴𝑂𝐶 + 𝑚∠𝐶𝑂𝐵 Two angles that form a linear pair are supplementary.
ao
bo
(s on a line)
“Linear pairs form supplementary angles.”
𝑎° + 𝑏° = 180 Given a sequence of 𝑛 consecutive adjacent angles whose interiors are all disjoint such that the angle formed by the first 𝑛 − 1 angles and the last angle are a linear pair, then the sum of all of the angle measures is 180°.
“Consecutive adjacent angles on a line sum to 180°.” 𝑎° + 𝑏° + 𝑐° + 𝑑° = 180
(∠𝑠 on a line)
The sum of the measures of all angles formed by three or more rays with the same vertex and whose interiors do not overlap is 360°.
“Angles at a point sum to 360°.”
(s at a point) 𝑚∠𝐴𝐵𝐶 + 𝑚∠𝐶𝐵𝐷 + 𝑚∠𝐷𝐵𝐴 = 360°
Lesson 6:
Solve for Unknown Angles—Angles and Lines at a Point
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6
M1
GEOMETRY
Facts (With Abbreviations Used in Grades 4–9)
Diagram/Example
The sum of the 3 angle measures of any triangle is 180°.
How to State as a Reason in an Exercise or a Proof
“The sum of the angle measures in a triangle is 180°.”
( sum of △)
𝑚∠𝐴 + 𝑚∠𝐵 + 𝑚∠𝐶 = 180°
When one angle of a triangle is a right angle, the sum of the measures of the other two angles is 90°.
“Acute angles in a right triangle sum to 90°.”
( sum of rt. △) 𝑚∠𝐴 = 90°; 𝑚∠𝐵 + 𝑚∠𝐶 = 90°
The sum of each exterior angle of a triangle is the sum of the measures of the opposite interior angles, or the remote interior angles.
“The exterior angle of a triangle equals the sum of the two opposite interior angles.”
(ext. of △) 𝑚∠𝐵𝐴𝐶 + 𝑚∠𝐴𝐵𝐶 = 𝑚∠𝐵𝐶𝐷
Base angles of an isosceles triangle are equal in measure.
“Base angles of an isosceles triangle are equal in measure.”
(base s of isos. △)
All angles in an equilateral triangle have equal measure.
“All angles in an equilateral triangle have equal measure."
(equilat. △)
Lesson 6:
Solve for Unknown Angles—Angles and Lines at a Point
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 6
M1
GEOMETRY
Facts (With Abbreviations Used in Grades 4–9)
Diagram/Example
If two parallel lines are intersected by a transversal, then corresponding angles are equal in measure. (corr. s, ̅̅̅̅ 𝐴𝐵 || ̅̅̅̅ 𝐶𝐷 ) If two lines are intersected by a transversal such that a pair of corresponding angles are equal in measure, then the lines are parallel.
How to State as a Reason in an Exercise or a Proof
“If parallel lines are cut by a transversal, then corresponding angles are equal in measure.”
(corr. s converse)
“If two lines are cut by a transversal such that a pair of corresponding angles are equal in measure, then the lines are parallel.”
If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are supplementary.
“If parallel lines are cut by a transversal, then interior angles on the same side are supplementary.”
(int. s, ̅̅̅̅ 𝐴𝐵 || ̅̅̅̅ 𝐶𝐷 ) If two lines are intersected by a transversal such that a pair of interior angles on the same side of the transversal are supplementary, then the lines are parallel.
“If two lines are cut by a transversal such that a pair of interior angles on the same side are supplementary, then the lines are parallel.”
(int. s converse) If two parallel lines are intersected by a transversal, then alternate interior angles are equal in measure. (alt. s, ̅̅̅̅ 𝐴𝐵 || ̅̅̅̅ 𝐶𝐷 ) If two lines are intersected by a transversal such that a pair of alternate interior angles are equal in measure, then the lines are parallel. (alt. s converse)
Lesson 6:
Solve for Unknown Angles—Angles and Lines at a Point
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M1-TE-1.3.0-07.2015
“If parallel lines are cut by a transversal, then alternate interior angles are equal in measure.”
“If two lines are cut by a transversal such that a pair of alternate interior angles are equal in measure, then the lines are parallel.”
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